Even if you cut down a tree to kill it, you haven't finished the task. Trees do not always die when cut; their elaborate root system continues to feed the plant, encouraging new sprouts to grow from the stump and roots. In order to eliminate this, you must kill the root system and stop new growth. Several chemical applications over a period of time slowly will force the entire system to die. Use caution to prevent killing other plants in the area.
Drill holes into the stump and exposed surface roots with a 1/2-inch drill bit. Remove debris from the surface of the stump and roots to prevent damaging your drill. Avoid the roots of the tree that cross over or get within a couple feet of other plant roots. The chemicals could transfer through the soil to other root systems if you get too close. Staying closer to the tree trunk usually solves this problem, a small risk of transfer remains.
Brush a concentrated glyphosate-based herbicide over the surface of the stump to coat it, and pour a small amount from the container into the drill holes. Do not dilute the mixture with water.
Pour small amounts of the concentrated glyphosate into the drill holes along the roots to fill them.
Trim away sprouts with pruning shears, but leave one or two uncut to monitor the root system. If the chemicals are working, the sprouts will die.
Watch for any new growth. Apply more chemicals if the sprouts do not die or if you see new sprouts popping up. Repeat applications every few months until you no longer see new growth.